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Random Auxetic Porous Materials from ParametricGrowth Processes

Jonàs Martínez

To cite this version:Jonàs Martínez. Random Auxetic Porous Materials from Parametric Growth Processes. 2020. �hal-02911788v1�

Random Auxetic Porous Materials from Parametric Growth Processes

Jonas Martınez

Universite de Lorraine, CNRS, Inria, LORIAAugust 4, 2020

Abstract

We introduce a computational approach to optimize ran-dom porous materials through parametric growth pro-cesses. We focus on the particular problem of minimiz-ing the Poisson’s ratio of a two-phase porous randommaterial, which results in an auxetic material. Initially,we perform a parametric optimization of the growthprocess. Afterward, the optimized growth process isused to directly generate an auxetic random material.Namely, the growth process intrinsically entails the for-mation of an auxetic material. Our approach enablesthe computation of large-scale auxetic random materi-als in commodity computers. We also provide numericalresults indicating that the computed auxetic materialshave close to isotropic linear elastic behavior.

Introduction

Auxetic structures are materials with negative Poisson’sratio: when stretched they expand perpendicularly tothe applied force [1, 2]. Auxetic materials find applica-tions in multiple fields thanks to their excellent shockabsorption, fracture toughness, or acoustical and vibra-tional absorption [3–8] among other reasons. A sub-stantial amount of research is devoted to the design ofauxetic mechanical materials [5, 9] that derive their phys-ical properties from the particular arrangement of theirsmall-scale geometry rather than from the material fromwhich they are made. Recent manufacturing technolo-gies can fabricate complex small-scale structures, andtherefore manufacture auxetic materials.

Random materials offer some advantages compared tothe more widespread periodic materials. In particular,they are more resilient to fabrication related symmetry-breaking imperfections [10], allow to compute the mate-rial geometry in an efficient and scalable way [11], andare capable to smoothly and seamlessly grade materialproperties and conform to any given surface [12]. Whilemost auxetic materials are defined by a repeating pe-riodic structure a distinct line of research is otherwiseinterested in random auxetic materials [13] since theyoffer certain advantages over periodic structures [14–16]. Auxetic polymeric foams [2, 17] were reported inthe 80s and are widely used in industrial applications.The most common process to obtain auxetic foams con-sists in compressing a conventional flexible cellular foamto force the cell ribs to buckle, producing a re-entrantstructure which is then heated to its softening tempera-ture [18, 19]. The geometry of cellular foams is usuallyidealized and modeled with Voronoi diagrams [20], and

some works studying auxetic foams start by modeling acompressed Voronoi diagram [21, 22]. Alternative meth-ods to produce random auxetic materials consider sheetswith random perforations [23] or random fiber networksthat can be produced by electrospinning [24].

A recent research direction tackles the minimization ofthe Poisson’s ratio of finite two-dimensional random net-works consisting of nodes connected by bonds. Differentoptimization approaches have been proposed, ideally as-suming that the elastic behavior of the network remainsisotropic during the iterative optimization process. Reidet al. [25] proposed to iteratively prune network bonds,further improved in [26] by modifying, in addition, theposition of the network nodes and the stiffness of thenetwork bonds. Hagh et al. [27] starts from a planartriangulation and iteratively removes bonds while avoid-ing the creation of reentrant polygons. Liu et al. [28]proposed an iterative adjustment of the bond stiffness.Pashine et al.[29] presented a network aging process thatdecreases the Poisson’s ratio.

We advocate for a different perspective on the problemof computing two-dimensional auxetic random materialsthat is computationally scalable. We consider a paramet-ric growth process that produces a porous material witha solid and void phase. The growth process is compactlydefined by a random point process and two parametricfunctions controlling the growth law. Our goal is to op-timize the parameters of these two functions in order tominimize the Poisson’s ratio of the resulting porous ma-terial. Importantly, this optimization is only carried outonce and allows afterward the immediate generation ofrandom auxetic materials through the process of growth,without requiring any further optimization process. Asshown in Figure 1, we enable the computation of large-scale random auxetic porous materials in commoditycomputers with close to isotropic behavior.

Results

Growth process

We consider a growth process [30] in which nuclei areborn at the same time. Each nucleus, a point x ∈ R2, isthe origin of a cell. Cells evolve according to a growthlaw and are forbidden to overlap. At the end of thegrowth, the union of all cells corresponds to the voidphase of the porous material, while its complement in R2

corresponds to the solid phase. We start by describinghow the nuclei are placed in space and how the law ofgrowth is defined.

1

Figure 1: From left to right: large-scale random auxetic porous material with increasing close-up views. The growthprocess has 43590 nuclei and the computation took 26 minutes on a laptop using a single CPU core.

A two-dimensional point process Φ ⊂ R2 defines thenuclei. We restrict ourselves to two different types ofpoint processes: a homogeneous Poisson point processwith intensity λ > 0, and a hard-core point processin which the points are forbidden to lie closer togetherthan a minimum distance D > 0 (e.g. the centers of ajammed packing of disks). In particular, we consider therandom sequential adsorption (RSA) model [31], weredisks with radius D

2 are iteratively added as long as theydo not overlap any previously added disk. The centerof the disks corresponds to the RSA point process. Thepoint process intensity of this RSA model is λ = 4A

πD2

, where A ≈ 0.547 is the area fraction were saturationoccurs [30].

Cells grow through a process of uniform scaling of acompact set S ⊂ R2 centered around each nucleus. Thegrowth ceases whenever and wherever a cell comes intocontact with another one (see Figure 2). This type ofgrowth process has been well studied in the literaturewhen S is a convex set and is also known as the Voronoigrowth model [32]. Our central idea is to allow S to benon-convex, enabling the formation through growth ofporous materials with more diverse shapes.

Φ

Figure 2: Given a point process Φ, the cells grow ac-cording to the uniform scaling of a Euclidean disk.

In addition, we restrict the growth process with whatwe call a maximum Euclidean radius of growth (see Fig-ure 3). Let Br ⊂ R2 be a closed Euclidean disk withradius r, and centered at the origin. Let x be a point inthe boundary of a growing cell. The growth ceases at xif x+Br intersects any other growing cell.

We start by giving a precise definition of S. Let S ⊂R2 be a compact star-shaped set with respect to theorigin O. That is, for any point x ∈ S, the line segment[x,O] is contained in S. The boundary of S can beparameterized in polar coordinates by a continuous andperiodic function ψS : [0, 2π] 7→ [rmin, rmax], where 0 <rmin ≤ rmax are the minimum and maximum Euclidean

Figure 3: Growth of two cells. The growth ceases at xsince the disk x+Br intersects another growing cell.

distance from O to the boundary of S. A descriptionof the method used to parameterize S is given in theMethods section. The distance induced by S from apoint p to a point x is [33]

dS(p, x) =‖x− p‖

ψS(∠(x− p)) , (1)

where ∠(·) is the angle of a vector with respect to the hor-izontal axis, and ‖·‖ is the Euclidean norm. Intuitively,dS(p, x) is the factor by which we have to uniformly scaleS, centered around p, such that x lies on its boundary.We always consider that p is a cell nucleus and x is anypoint of the cell emanating from p.

The maximum Euclidean radius of growth is parameter-ized by a star-shaped set S∗. We consider a polar func-tion ψS∗ : [0, 2π] 7→ [r∗min, r

∗max], where 0 < r∗min ≤ r∗max

are the minimum and maximum Euclidean radius ofgrowth. Let x be a point in the boundary of a growingcell emanating from p ∈ Φ. The growth ceases at x ifx+BψS∗ (∠(x−p)) intersects any other growing cell.

The minimal distance between two points in a Poissonpoint process can be arbitrarily small. Thus, a cell canbe empty as its growth was forbidden through S∗ fromthe very beginning of the growth process. For a hardcorepoint process, the minimal distance between two pointsis at least D and it suffices to enforce D > r∗max in orderto guarantee that all cells are allowed to grow.

It is challenging to formulate a continuous growthlaw [34] in our setting since both S and S∗ are not neces-sarily convex, and S∗ imposes a distinct local constrainton the growth process. We instead resort to a discretegrowth process formulation, where the cells grow in dis-crete steps. Therefore our growth process defines a dis-crete random set. In particular, we consider an integerlattice LW = Z2∩W where W is a bounded subset of R2.

2

Let ΦW = Φ ∩W = (p1, . . . , pn) be the subset of nucleiin W . The set of lattice points belonging to the cells isgiven by the union CW =

⋃ni=1 C

iW ⊂ LW , where CiW

corresponds to the lattice cell points associated with thenuclei pi. Lattice points are incrementally added to CWthroughout the discrete growth process. Let S1 ⊂ R2 bethe unit square centered at the origin, and let ⊕ denotethe Minkowski sum operator. We consider that the voidphase of the porous material is given by CW ⊕S1, whilethe solid one is given by (LW \CW )⊕S1. If W is a rect-angle, both phases can be compactly represented with abinary digital image. In all illustrations the void phase iscolored in white and the solid phase in black. Figures 4and 5 show an overview of the growth process.

S

Figure 4: Growth process given by the point process ofFigure 2 and the uniform scaling of S at the left. In thisexample S∗ is a Euclidean disk.

S∗

Figure 5: Growth process given by the point process ofFigure 2, the uniform scaling of S in Figure 4, and themaximum Euclidean radius of growth S∗ at the left.

In the following, we describe in more detail Algorithm 1that simulates the process of discrete growth. An or-dered set Q contains tuples {x, p} where the first ele-ment is a lattice point of x ∈ LW and the second one isa nucleus point of p ∈ ΦW . Q is ordered according tothe distance dS(p, x) and is used to simulate the discretegrowth process over LW . The tuples in Q identify thepoints where cell growth can potentially take place. Thegrowth process is divided in two parts, the nucleationphase and the growth phase.

Nucleation Before the growth starts, for all pi ∈ ΦWwe insert in Q the tuple {xi, pi}, where xi ∈ LW is theclosest lattice point to pi. LW should be sufficientlydense so no two points in ΦW are closest to the samelattice point in LW .

Growth The discrete growth process is simulated asfollows. Let {x, pi} be the tuple in Q with the smallestdistance dS(pi, x) among all tuples.

- (Line 8) First, we remove {x, pi} from Q.

- (Lines 9–12) Second, if x does not already belong toany cell and the Euclidean disk BψS∗ (∠(x−pi)) does

not contain any other cell point we proceed to addx in CiW . That is, the lattice point x now belongsto the cell emanating from pi.

- (Lines 13–14) Finally, future candidate points ofgrowth around a discrete local neighborhood of pare inserted Q. More precisely, we insert in Q thefour tuples {xn, pi}, where xn is a lattice point inthe 4−connected neighborhood of x in LW .

The above is done iteratively until Q is empty.

Periodic boundary conditions Optionally, whenW is a rectangle periodic boundary conditions can beeasily imposed by considering that LW is a periodic lat-tice as well. If Φ is an RSA point process, the randomadsorption of disks has to be performed considering aperiodic domain. In this work, periodic boundary con-ditions are used to generate porous materials that canbe simulated with periodic homogenization.

Algorithm 1 discreteGrowthProcess(LW , ΦW , S, S∗)1: CiW ← ∅ for all CiW ∈ CW2: for pi ∈ ΦW do . Nucleation3: Let xi ∈ LW be the closest lattice point to pi4: Insert in Q the tuple {xi, pi}5: end for6: while Q 6= ∅ do . Growth7: Let {x, pi} ∈ Q with smallest distance dS(pi, x)8: Remove {x, pi} from Q9: if x /∈ CW then

10: N ← LW ∩(x+BψS∗ (∠(x−pi))

)11: if (y /∈ CW ) ∨ (y ∈ CiW ) for all y ∈ N then12: CiW ← CiW ∪ {x}13: for xn ∈ neighborhood of x in LW do14: if xn /∈ CW then15: Insert in Q the tuple {xn, pi}16: end if17: end for18: end if19: end if20: end while

Cell regularization In our discrete setting, connect-edness is given for the lattice LW with respect to a 4-connected neighborhood. Algorithm 1 is guaranteed toproduce discrete cells CiW that are connected sets. How-ever, when S∗ is not convex it may occur that a cellhas a genus higher than zero (i.e. has “holes”). Thisleads to a porous material solid phase composed of morethan one connected component, which is undesirable.To resolve this we fill any hole that a cell may have. For-mally, we consider the complement of the unboundedcomponent of the complement of a cell (see Figure 6).We denominate this last step cell regularization

Stochastic homogenization

We study the elastic behavior of the random porous ma-terials given by our growth process. We restrict ouranalysis to linear elasticity which is useful to model the

3

S S∗ Regularized

Figure 6: Left: S and S∗ governing the growth processof two nuclei. Middle: evolution of the growth, the holesarise due to the high anisotropy of both S and S∗, andthe particular arrangement of the two nuclei. Right: thetwo cell holes are simply filled.

behavior of materials under small deformations. In par-ticular, we consider the stochastic homogenization [35]of the random porous material in order to derive itsaverage linear elastic constitutive behavior.

Hooke’s law represents the material behavior of elasticmaterials with the linear relation σ = Cε, where σ isthe stress, ε is the strain, and C is the elasticity ten-sor (a symmetric and positive definite matrix), in Voigtnotation [36] σ1σ2

σ12

=

c11 c12 c13c12 c22 c23c13 c23 c33

ε1ε2

2ε12

(2)

Periodic homogenization [37] seeks to find the elastic-ity tensor characterizing a periodic composite materialdefined from a periodic cell, as the length of the celltends to zero. In the periodic case, the homogeniza-tion is defined on a finite domain (periodic cell) whileon the random case is defined on the whole space R2,and cannot be reduced to a problem posed on a finitedomain [38].

We approximate the homogenized elasticity tensor coef-ficients of a random porous material, by means of theso called “cut-off” techniques. Consider a periodizedcut [0, s]

2(a square) of a stationary random material

and its corresponding homogenized elasticity tensor Cs.It was shown in [38] that lims→∞ Cs converges, almostsurely. Nevertheless, s needs to be bounded for practicalpurposes of computation.

In our discrete setting, s is a positive integer value thatcorresponds to the pixel resolution s× s of the porousmaterial image. In all results, we consider a constantD = 80 for the RSA point process, and λ = 4A

πD2 ≈0.000 11 for the Poisson point process. Therefore, bothtested point processes have similar intensity.

Let E[·] be the expectation of a random variable. Weestimate the value of E[Cs] with the average of k in-dependent realizations of the random porous materialgiving (Cs)i, 1 < i ≤ k

E[Cs] ≈ Cs =1

k

k∑i=1

(Cs)i (3)

This is an approximation (Monte Carlo method) fre-quently used in the context of stochastic homogeniza-tion [35]. Let std(Cs) be the standard deviation of thek independent realizations (Cs)i. A range of plausible

values for E[Cs] is(Cs ± 1.96 std(C

s)√k

)for a confidence

interval with probability 95% (see Figure 7).

In order the evaluate the accuracy of the stochastic ho-mogenization [39] we examine the statistical fluctuationsof the material porosity p ∈ [0, 1] (the area fraction ofthe void phase) for a given k and s. In particular, we

consider the coefficient of variation cv(p) = std(p)p that

indicates the extent of variability of the porosity in rela-tion to the mean. Increasing the value of k and s mostlikely decreases cv(p). Following [40] we consider thatthe approximation is sufficiently precise if cv(p) is belowa maximum predefined one. In all the following resultswe always consider k = 60 realizations.

−→

i = 1 i = 2 i = 120

0 15 30 45 60 75 90 105 120

i

0.001

0.002c11

c12

c13

c22

c23

c33

Figure 7: Approximation of the elasticity tensor of therandom porous material of Figure 5 in a square with sizes = 1200, and a total of 120 independent realizations.Left: Some realizations. Right: mean and confidenceinterval of each tensor component. As the number ofrealizations i increases, the mean stabilizes and the con-fidence interval becomes narrower. The coefficient ofvariation of porosity is cv(p) = 0.004 09.

Isotropic elasticity

Since our goal is to minimize the Poisson’s coefficient ofan isotropic material, we give here a brief presentationof how we approximate the isotropic elastic behavior ofa random material elasticity tensor, given its averageelasticity tensor Cs.

An isotropic material has an elasticity tensor Ciso withonly two independent components ciso12 and ciso12

Ciso =

ciso11 ciso12 0ciso12 ciso11 0

0 0ciso11 −c

iso12

2

(4)

The relation between the tensor components ciso11 , ciso12

and the Young’s moduli E > 0 and the Poisson’s ratiov ∈ [−1, 1] is given by

E =(ciso11 )

2 − (ciso12 )2

ciso11

v =ciso12

ciso11

(5)

4

In practice, it seldomly occurs that an elasticity tensorCs is perfectly isotropic. Thus, we seek to determine itsclosest isotropic tensor [41]. We consider the standardFrobenius norm for a 3× 3 matrix A

‖A‖F =

√√√√ 3∑i=1

3∑j=1

|aij |2 (6)

and minimize the Frobenius norm between Ciso and theaverage tensor Cs

minciso11 ,c

iso12

∥∥Cs − Ciso∥∥F

(7)

We provide in the Methods section the closed-form ex-pression of ciso11 , c

iso12 minimizing Equation (7).

We also need a measure of the deviation of Cs from per-fect isotropy. We consider the typically used normalizedmeasure δiso [15, 42]

δiso =

∥∥Cs − Ciso∥∥F∥∥Cs∥∥

F

≥ 0 (8)

where δiso = 0 indicates perfect isotropy, and increasingvalues indicate divergence from isotropy.

It is known that periodic structures in R2 having three-fold rotational symmetry (invariant to rotations of 2π

3 )lead to an isotropic linear elastic behavior [43]. Wehypothesize that a resembling property holds in our in-stance of random growth.

A two-dimensional point process Φ is stationary if Φ+xhas the same probability distribution than Φ, for allx ∈ R2. Φ is isotropic if R(a)Φ has the same probabilitydistribution than Φ, for all rotations R(a) of angle a ∈R around the origin. Both the Poisson point processand the RSA point process are stationary and isotropicpoint processes [30, 44]. The notion of stationarity andisotropy extends to general random sets.

Let us consider a rotation of R( 2π3 n), for n ∈ Z. The

probability distribution of Φ is invariant to a rotationR( 2π3 n). Moreover, if S and S∗ are three-fold symmetricthen R( 2π

3 n)S = S (same for S∗). Given a point p ∈ Φthen we have

R(2π

3n)(p+S) = R(

2π

3n)p+R(

2π

3n)S = R(

2π

3n)p+S

(9)i.e., the rotation of S centered at p around the originkeeps S and S∗ invariant. We infer that the growthprocess is stationary and invariant to rotations R( 2π

3 n)which leads us to the following hypothesis.

Hypothesis 1. Let Φ be a stationary and isotropicpoint process. Let S and S∗ be three-fold rotational sym-metric. For s→∞ the discrete growth process describedin this article gives a porous material with δiso close tozero. That is, the resulting porous material has close toisotropic linear elastic behavior.

We have numerically evaluated Hypothesis 1 for a largeset of porous materials with different sets S and S∗, andincreasing size s (see Figure 8). There is strong evidencethat as s increases the average value of δiso decreases andit appears to tend towards a value close to zero.

In all the subsequent optimization results consideringthree-fold rotational symmetric sets S and S∗ we con-sider a fixed square size of s = 1200. This size gaveamong all the tests performed in Figure 8 a maximumcoefficient of variation of porosity of cv(p) = 0.035 19 forthe Poisson point process, and of cv(p) = 0.011 18 forthe RSA point process.

We also evaluate the impact of the cell regularizationover the porosity. More precisely, we measure the rela-tive amount of cell holes that are filled during the cellregularization. Let p be the porosity of the porous ma-terial before performing the cell regularization. We con-

sider the relative measure ζp,p =∣∣∣ p−pp ∣∣∣ ≥ 0. For all the

results in Figure 8 the maximum value of ζp,p is 0.061 57and the mean is 0.004 33 for the Poisson point process,and maximum 0.019 48 and mean 0.000 41 for the RSApoint process. This indicates that the effect of the cellregularization on the porosity is almost negligible, andis particularly small for the RSA point process.

Parametric optimization

By this point, we have all the ingredients to formu-late the parametric optimization of the growth process.Let Cs(ψS , ψS∗) be the average elasticity tensor of aporous material parameterized by the functions ψS (star-shaped distance) and ψS∗ (maximum Euclidean radiusof growth). Let F : M 7→ R be the objective function,where M is a 3× 3 symmetric matrix. Our objective isto minimize

arg minψS ,ψS∗

F(Cs(ψS , ψS∗)) =ciso12

ciso11

(10)

where ciso11 , ciso12 are derived from Cs(ψS , ψS∗) considering

Equation (15). In order to obtain a physical behaviorclose to isotropic elasticity we constrain S and S∗ tobe three-fold rotational symmetric. More details of par-ticular choice of optimization method are given in theMethods section.

Figures 9 to 11 provide minimization results of thePoisson’s ratio with an increasing optimization domainof ψS∗ , and a constant optimization domain of ψS ∈[0.05, 1.0].

Discussion

We have presented a novel computational approachbased on parametric growth processes to minimize thePoisson’s ratio of random porous materials. The resultsattain a negative Poisson’s ratio while providing a care-ful numerical analysis showing close to isotropic behav-ior. We observe that increasing the number of parame-ters of ψS and ψS∗ and their domain further improvesthe minimization. We recommend the use of a hardcore

5

RSA point process Poisson point process

300 600 9001200

Size s

0.00

0.05

0.10

δ iso

300 600 9001200

Size s

0.00

0.05

0.10

δ iso

300 600 9001200

Size s

0.00

0.05

0.10

0.15

c v(p

)

300 600 9001200

Size s

0.00

0.05

0.10

0.15

c v(p

)

Figure 8: Analysis of the deviation of isotropy δiso and coefficient of variation of the porosity cv(p) for a largenumber of porous materials, and two different point processes (left and right columns). S and S∗ are three-foldrotational symmetric. The colored boxes encompass the interquartile range of values of δiso. We consider 4 differentsizes s, going from 300 to 1200 For each point process and size we enumerate 300 different random radial spans inthe range ψS ∈ [0.05, 1.0] and ψS∗ ∈ [4.0, 40.0], and 8 different equally spaced angles in [0, 2π3 ] (three-fold symmetry)considering a total of 2 · 4 · 300 = 2400 different porous materials. In general, we observe a clear trend of the averagevalue of δiso and cv(p) getting closer to zero as the size s of the domain increases. We also notice lower averagevalues with the RSA point process compared to the Poisson point process.

0 50 100 150Number of calls

−1.0

−0.5

0.0

0.5

1.0

v

v = −0.179 58δiso = 0.024 82

Poisson S

0 50 100 150Number of calls

−1.0

−0.5

0.0

0.5

1.0

v

v = −0.322 03δiso = 0.031 54

RSA S

Figure 9: Optimization with constant ψS∗ = 4.0.

point process instead of a Poisson point process to reachlower Poisson’s ratio and lower variance of the materialproperties in a finite extent (see Figure 8).

We distinguish two clear trends from the optimizationresults. First, the RSA point process is able to achievelower Poisson’s ratio v compared to the Poisson pointprocess. Second, increasing the domain of function ψS∗

leads to lower Poisson’s ratio, since we expand the spaceof achievable porous materials.

With a parametric growth process, it is possible to gener-ate heterogeneous materials by simply considering thatboth funcitions ψS and ψS∗ are varying in the space(see Figure 2). One of the main advantages of our ap-

proach over periodic structures is that the transitionsbetween different material properties do not need to behandled [45], as they are intrinsically captured by pa-rameters governing the growth process.

Our approach also opens up different opportunities forfuture work. For instance, to extend it to the 3D caseby considering a 3D growth process and adapting theparameterization of S and S∗. This may be of interestfor the inverse design of random 3D porous materials,e.g. random foams are of interest for acoustic absorp-tion [46] or tissue engineering [47]. Moreover, our ap-proach may open the door to other applications besidesthe minimization of the Poisson’s ratio of linearly elas-tic materials. For example, to tackle the inverse designand optimization of porous materials [12, 48] using ourgrowth process. In most cases, it is sufficient to replacethe objective function of Equation (10) with another oneof interest.

Methods

Parameterization of the star-shaped set S Fol-lowing [33] we parameterize S with a few known values(αi, li)i>0 of the function ψS(αi) = li, and we call themradial spans. For simplicity we always consider m > 0equally spaced radial spans separated by an angle ofαm = m

2π , and starting at an angle of zero α1 = 0. Letαjk ∈ [0, 2π] be an angle that we seek to interpolate, ly-ing between (αj , lj) and (αk, lk), such that k = bαjkαmcand j = (k + 1) mod m. The relative position t of αjkbetween αk (t = 0) and αj (t = 1) is given by the frac-tional part t = αjkαm − bαjkαmc ∈ [0, 1). Then, weinterpolate the value of ψS(αjk) by considering the fol-lowing cubic Hermite spline on the unit interval

ψS(αjk) = (2t3 − 3t2 + 1)lk + (−2t3 + 3t2)lj (11)

6

0 50 100 150Number of calls

−1.0

−0.5

0.0

0.5

1.0

v

S∗

v = −0.518 48δiso = 0.048 23

Poisson S

0 50 100 150Number of calls

−1.0

−0.5

0.0

0.5

1.0

v

S∗

v = −0.710 22δiso = 0.015 15

RSA S

Figure 10: Optimization with variable ψS∗ ∈ [4.0, 15.0]

0 50 100 150Number of calls

−1.0

−0.5

0.0

0.5

1.0

v

S∗

v = −0.572 39δiso = 0.013 47

Poisson S

0 50 100 150Number of calls

−1.0

−0.5

0.0

0.5

1.0

v

S∗

v = −0.768 46δiso = 0.014 37

RSA S

Figure 11: Optimization with variable ψS∗ ∈ [4.0, 40.0]

Figure 12: Spatially varying growth process with two different growth functions. Left: Input heterogeneous field,the darker regions correspond to the optimized ψS in Figure 9, while the rest correspond to a growth according tothe Euclidean disk.

The above interpolation ensures that ψS(αjk) lies inbetween lk and lj by setting the starting and endingtangents of the spline to zero. In addition, we can eas-ily impose rotational symmetries to the interpolation.An illustration of the interpolation procedure is shownin Figure 13.

(αj , lj)O

(αk, lk)

αm

Figure 13: Star-shaped set parameterization. Fiveequally spaced radial spans and the interpolation result.Left: imposing three-fold rotational symmetry.

Numerical homogenization We compute the ho-mogenized elasticity tensor Cs with a publicly availablenumerical homogenization method based on the finiteelement method [49]. The input is an image with values

identifying each distinct phase of the periodic material.We impose periodic boundary conditions on the growthprocess in order to produce a suitable input image. Thesolid phase is composed of a linear elastic isotropic ma-terial with unit Young’s modulus E = 1.0 and Poisson’sratio v = 0.3.

Parametric optimization method The input offunction F is expensive to evaluate, has no closed form,and it involves a numerical Montecarlo approximation.In this case, Bayesian optimization [50] is often a goodchoice to tackle the optimization. In particular, we haveused Bayesian optimization using Gaussian processes asimplemented in the Python library scikit-optimize [51].Both functions ψS and ψS∗ are parameterized withm = 8 radial spans. We set the first radial span ofψS to a constant value of 1, removing an unnecessarydegree of freedom. For the Bayesian optimization, weset a maximum number of 150 evaluations of functionF, and 50 random initial guesses.

7

Derivation of ciso11 and ciso12 For the three-dimensionalcase, the minimizing values of ciso11 , c

iso12 were derived, for

instance, in [52]. We were not able to find any previ-ous work concerning the simpler two-dimensional case.Nevertheless, the values minimizing Equation (7) canbe derived by simple calculus. Consider the function fwith two arguments ciso11 and ciso12

(12)

f =∥∥C − Ciso∥∥2

F

= (c11 − ciso11 )2 + 2(c12 − ciso12 )2 + (c22 − ciso11 )2

+ 2c213 + 2c223 +

(c33 −

ciso11 − ciso12

2

)2

whose first-order partial derivatives are

∂f

∂ciso11

= −c33 − 2(c22 − ciso11 ) +ciso11 − ciso12

2− 2(c11 − ciso11 )

∂f

∂ciso12

= c33 − 4(c12 − ciso12 )− ciso11 − ciso12

2(13)

The Hessian matrix H of f

H =

∂2f∂(ciso11 )2

∂2f∂ciso11 c

iso12

∂2f∂ciso12 c

iso11

∂2f∂(ciso12 )2

=

[92 − 1

2− 1

292

](14)

is positive definite everywhere and it follows that f is

a convex function. Since ∂2f∂(ciso11 )2

= ∂2f∂(ciso12 )2

> 0 the

function f has a global minimum at

ciso11 =1

20(9(c11 + c22) + 2c12 + 4c33)

ciso12 =1

20(c11 + c22 + 18c12 − 4c33)

(15)

found by solving the system of linear equations fromEquation (13) being equal to zero.

Code availability

An open source implementation of our approach can befound at https:\\future-url-code. We provide anscript to automate the generation of all paper results andfigures, apart from the manually done Figure 3.

Data availability

All the results of this article can be downloaded fromhttps:\\future-url-data.

Acknowledgments

This work was partly supported by ANR MuFFin (ANR-17-CE10-0002).

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